4 research outputs found
Regular and First Order List Functions
We define two classes of functions, called regular (respectively, first-order) list functions, which manipulate objects such as lists, lists of lists, pairs of lists, lists of pairs of lists, etc. The definition is in the style of regular expressions: the functions are constructed by starting with some basic functions (e.g. projections from pairs, or head and tail operations on lists) and putting them together using four combinators (most importantly, composition of functions). Our main results are that first-order list functions are exactly the same as first-order transductions, under a suitable encoding of the inputs; and the regular list functions are exactly the same as MSO-transductions
The semaphore codes attached to a Turing machine via resets and their various limits
We introduce semaphore codes associated to a Turing machine via resets.
Semaphore codes provide an approximation theory for resets. In this paper we
generalize the set-up of our previous paper "Random walks on semaphore codes
and delay de Bruijn semigroups" to the infinite case by taking the profinite
limit of -resets to obtain -resets. We mention how this opens new
avenues to attack the P versus NP problem.Comment: 28 pages; Sections 3-6 appeared in a previous version of
arXiv:1509.03383 as Sections 9-12 (the split of the previous paper was
suggested by the journal); Sections 1-2 and 7 are ne
Random walks on semaphore codes and delay de Bruijn semigroups
We develop a new approach to random walks on de Bruijn graphs over the
alphabet through right congruences on , defined using the natural
right action of . A major role is played by special right congruences,
which correspond to semaphore codes and allow an easier computation of the
hitting time. We show how right congruences can be approximated by special
right congruences.Comment: 34 pages; 10 figures; as requested by the journal, the previous
version of this paper was divided into two; this version contains Sections
1-8 of version 1; Sections 9-12 will appear as a separate paper with extra
material adde
Asymptotic, Algorithmic and Geometric Aspects of Groups Generated by Automata
This dissertation is devoted to various aspects of groups generated by automata. We
study particular classes and examples of such groups from different points of view. It
consists of four main parts.
In the first part we study Sushchansky p-groups introduced in 1979 by
Sushchansky in "Periodic permutation p-groups and the unrestricted Burnside
problem". These groups represent one of the earliest examples of Burnside groups
and, at the same time, show the potential of the class of groups generated by automata
to contain groups with extraordinary properties. The original definition is translated
into the language of automata. The original actions of Sushchansky groups on p-
ary tree are not level-transitive and we describe their orbit trees. This allows us
to simplify the definition and prove that these groups admit faithful level-transitive
actions on the same tree. Certain branch structures in their self-similar closures
are established. We provide the connection with so-called G groups introduced by
Bartholdi, Grigorchuk and Suninc in "Branch groups" that shows that all Sushchansky
groups have intermediate growth and allows us to obtain an upper bound on their
period growth functions.
The second part is devoted to the opposite question of realization of known
groups as groups generated by automata. We construct a family of automata with n states, n greater than or equal to 4, acting on a rooted binary tree and generating the free products of
cyclic groups of order 2.
The iterated monodromy group IMG(z2+i) of the self-map of the complex plain
z -> z2 + i is the central object of the third part of dissertation. This group acts
faithfully on the binary rooted tree and is generated by 4-state automaton. We provide
a self-similar measure for this group giving alternative proof of its amenability. We
also compute an L-presentation for IMG(z2+i) and provide calculations related to the
spectrum of the Markov operator on the Schreier graph of the action of IMG(z2 + i)
on the orbit of a point on the boundary of the binary rooted tree.
Finally, the last part is discussing the package AutomGrp for GAP system developed
jointly by the author and Yevgen Muntyan. This is a very useful tool for studying
the groups generated by automata from the computational point of view. Main
functionality and applications are provided